Deriving S = k ln Ω

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  • Derlvlng S = k Inn To the Editor:

    We wish to comment upon Melvyn P. Melrose's article "Deriving S = k InR" [1983,60,106]. This remarkable article lucidly presents the subject matter in a most convincing manner even to beginners in chemistry and physics. I t is intriguing to us that the author in the development of his relations has considered the first law of thermodynamics, dE = TdS - PdV, with only two variables. However, for the sake of generality, in the absence of various external forces such as electric, magnetic, gravitational, tensions, i t should have been considered that

    dE = TdS -PdV + &Vj (1) so that

    d~ = ( a ~ ~ a s ) , , ~ d s + ( ~ E I ~ v ) ~ ~ ~ V + Z (~EI~N,)S,V w (2) to be in a position to compare with d~ = (a~/a~nn),dlna + ( ~ E I ~ v ) , , ~ d v + ( a~ la~ , ) , , , , dn: (3) As a result, it becomes necessary to prove that

    pi = (aElaNi)s,V = (aEIaNA,," (4) so that the derivation becomes completely general. In the Melrose's article, eqs 2 and 20 should have been

    and P = -(~EI~v),,, (6)

    730 Journal of Chemical Education

  • These relations could be derived in the same fashion with the additional constraint that the total number of micro- states were held constant. Relation 4 has been alreadv de- fined but not proved ( I ) . Equation 4 can be derived in the same fashionas that of Melrose (his ea 20) with the addition- al constraint that V is constant. Fo; verification, we have reneated the derivation and the results need not be repro- diced here.

    The entropies that are calculated bv the methods of classi- cal thermodynamics and sratisriral mechanics are com- oared. and the differences if any are ascrhed to the disorder br the glassiness of the crystal, the entropy of mixing a t absolute zero. This difference is also identical with the dif- ference in the theory of classical thermodynamics and the experiment (2). The author's comment on the statement that the "textbooks do not prove that the entropy as defined in statistical thermodynamics is the same as the entropy in classical thermodynamics" seems to be unfortunate, since these values agree on the same footing.

    Classical thermodvnamics is developed based onlv on re- versihle or equilibrium processes. S~atistiral merhanical methods are ao~l i rable only for systems that have attained complete and hjmamic equilibrium. Therefore, the question of thermodvnamic eauilihrium beina d i l ' f~~en t from statisti- cal equilibiium sho&d not arise. Here again, the author's assumption of statistical and thermodynamical equilibrium being identiral also seems to be unneressary.

    The third comment that we would like to make is on the statement that "direct applications of S = klnn are restrict- ed to ideal systems and thus do not show that the formula is generally valid". (1) For the development of statistical mechanics we consider iden-

    tical independent localized systems which is an ideal model and derive the famous Boltzmann distribution expression. From this, energy is calculated and converted to heat capacities at constant volume (C,). This statistically derived expression for C, agrees very well to some simple solids as has heen done by Debye (3).

    (2) When we consider an assembly of identical independent nonlo- ealized systems, it is nothing hut an ideal gas. However, the statistical theom develooed and extended to ratio of s~ecific heats at ronstanr prcrsure and volume agrer wry in rase of the erpcrimrnrnl vnlues for inert rnunuarumir gases like neon. argon, krypton, and xenon (31.

    Our points (1) and (2) contradict the author's statement.

    Literature Cited

    Ramachandra H. Balundal and Telral M. Aminabhavl

    To the Editor:

    Aminabhavi and Balundei's main noint is correct. Mv treatment would have been complete i i 1 had considered thk possibility that my subsystems A and B were open systems. They are quite right to point this out, and I think they would be iustified in nresentine the com~le te analvnis. i.e., in deriv- in i the statistical analog for r,, just as I found the statistical analog for P .

    However, their other observations are either based on misunderstandings of my paper, or are incorrect in them- selves. (1) "Texthooks do not prove that the entropy as defined in statisti-

    cal mechanics is the same as the entropy in classical thermodyn- amics." My statement referred to ded~ctrue proof. I regard the

    agreement between statistical entropies and calorimetric (third law) entropies as inductive evidence, which must be regarded as inconclusive especially when discrepancies that arise between the two quantities are interpreted statistically!

    (2) The assumption that statistical and thermodynamic equilibri- um are the same is not self-evident, and it is necessary to admit this at the beginning. In statistical mechanics a system is in equilibrium when every microstate (with the same E and N) occurs with the same orobabilitv. It is not self-evident that this is also a system in which intensive variables (T, P, a,) do not change with time.

    (3) "Direct applications of S = klnn are restricted to ideal sys- tems.. . ." By "direct", I meant: computing 0 directly, as the number of microstates contained in the most probable distribu- tion of energy among particles. It is clear that for this calcula- tion to he feasible, the particles have to be independent; that is, the system has to be ideal.

    I do not regard the method of the canonical ensemble, in which the particles are replaced by large subsystems, (which may be nonideal systems) as a direct application of S = klnQ, since entropy of each subsystem is given by S = -kxPJnPi. The kev auantitv is the molar (not the molecular) partition function. ~ h u s , iregard ~ e b ~ e ' s treatment of the ;&id as an a~nlication of the canonical ensemble. The other examnles that they mention (ideal gases) would seem to confirmAmy statement, not contradict it.

    M i n g k 9 ; , : ? : n - Strand, London WCZR 2LS. England

    Asslgnlng Absolute Conflguratlon To the Editor:

    The "New Gimmick for Assigning Absolute Configura- tion" a t a chiral center by Ayorinde [1983,60,928] is similar to the method for assignment of R and S configuration which I have been teachine to introductorv oreanicchemistrv stu-

    " " - ~ ~~ 9 ~~~~

    dents for some years. However, I feel that Ayorinde's meth- od has a feature that has the notential for creatine consider- able confusion for some s t ~ d ~ n t s . Step 3 of ~ ~ o r i t d e ' s meth- od makes use of a structure I, which has the same two- dimensional arrangement of substituents as a Fischer projection. However, this structure requires a different con- vention for vertical and horizontal bonds from that implied by the Fischer projection. To my mind, despite the dotted bonds in the structure, the use of two different conventions for the same two-dimensional arrangement of substituents is likely to lead to problems for the introductory (and particu- larly the less able) student.

    My own approach is to introduce the Fischer projection and then carefully establish (using molecular models) that the interchanae of anv air of substituents leads to an inver- ". sion of configkation a t the chiral center. This then allows use of what is a modification of the nrocedure described bv .4wrinde tor the asoignmrnt o i configuration in cases when the lowest priority sut,stiturnt is not at the ton or lr,rtom of the Fische;

    Modify steps 3 and 4 of Ayorinde's method as follows: insert structure

    Step 3. Switch the group of lowest priority (d) with the substitu- ent at the top (or bottom) of the Fiseher projection. This leads to inversion of configuration; i.e., B is the enantiomer of A.

    Volume 64 Number 8 August 1987 731